On the probability that all decision rules select the same winner

Journal of Mathematical Economics 33 Ž2000. 183–207 www.elsevier.comrlocaterjmateco On the probability that all decision rules select the same winner V. Merlin a,) , M. Tataru b,c , F. Valognes d,e a GEMMA-CREME and CNRS, MRSH-SH230, UniÕersite´ de Caen, Esplanade de la Paix, F-14032 Caen Cedex, France b Department of Mathematics, College of New Jersey, Ewing, NJ, USA c Department of Mathematics, Northwestern UniÕersity, 2033 Sheridan Road, EÕanston, IL 60208-2730, USA d GEMMA-CREME, MRSH-SH231, UniÕersite´ de Caen, Esplanade de la Paix, F-14032 Caen Cedex, France e Department of Economics, UniÕersity of Namur, 8 Rempart de la Vierge, B-5000 Namur, Belgium Received 5 November 1997; received in revised form 2 April 1999; accepted 9 April 1999 Abstract Social choice literature proposes several decision processes to achieve a social compromise. We know that different procedures may yield different solutions. However, no study has been done to estimate the probability that they simultaneously pick the same outcome. We partially answer this question for three-candidate elections. Let P U be the probability that almost all the voting rules: the Condorcet procedures, the scoring rules and the runoff methods select simultaneously the same winner. We provide new techniques for the computation of the occurrence of voting outcomes when the number of voters approaches infinity and each voter selects independently a linear preference order on the candidates according to a uniform probability distribution. With these assumptions, the exact value of P U turns out to be 0.50116. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: D71 Keywords: Condorcet; Decision making; Probability; Schlafli; ¨ Voting rules ) Corresponding author. Tel.: q33-2-3156-6249; fax: q33-2-3156-6248; E-mail: [email protected] 0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 1 2 - 9 184 V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 1. Introduction Throughout history, the question of the choice of the best voting procedure always aroused the curiosity of philosophers and politicians in societies where democratic processes have been used to select a social outcome. Without being exhaustive, we could mention several Greek and Roman authors, all the middle age literature about the election of a Pope and of course the debate between Borda and Condorcet in the Academie Royale des Sciences, just before the French ´ Revolution. Arrow Ž1963. analyzed this question in a mathematically rigorous and fruitful framework, which led him to the famous Impossibility Theorem: ‘‘There is no aggregation procedure, except dictatorship, which can satisfy simultaneously three a priori benign axioms.’’ In some sense, this result means that the perfect democratic decision process does not exist. This rather negative conclusion paradoxically favored the development of comparative analysis of aggregation procedures. The first reason is that Arrow’s formalization provides a unified framework in which voting rules can be rigorously compared. Secondly, from a pragmatic point of view, the Impossibility Theorem does not exclude any rule from being a good one, although all procedures suffer imperfections. Hence, a social choice researcher can easily find arguments to support the method she prefers. The literature on the properties and defaults of most common voting rules exploded in the last 3 decades. 1 However, these works would be in vain if all the rules would select the same outcome in almost all the situations. The choice of a procedure only matters if, by changing the decision process, the social outcome may radically differ from the previous one. Although important, this issue has not received a lot of attention from Social Choice researchers until the beginning of the 1980s. Since that time, several authors tried to evaluate the discrepancies among voting rules, from a qualitative and a quantitative point of view. The most spectacular analysis in this category is probably due to Saari Ž1992., who generalizes the results of Fishburn Ž1981.. Consider a finite group of individuals, N s  1, . . . n4 , who has to rank collectively several alternatives a1 , . . . , a m Ž m G 3 is finite.. We assume that each individual is able to rank all these options from her most to her least preferred without tie, i.e., the individual preferences are represented by linear orderings. 2 A n-tuple of linear orderings, one for each individual, is called a preference profile. Next, Saari suggests that the society should use a scoring rule or positional rule in order to rank the alternatives. Let w s Ž w 1 ,w 2 , . . . wp , . . . wm . be a m-dimensional scoring Õector, 1 As a sample of books about this subject, we recommend the works of Dummett Ž1984., Nurmi Ž1988. and Saari Ž1994.. 2 Recall, a linear ordering is a complete, transitive and antisymmetric binary relation. V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 185 where w1 s 1, wm s 0 and wp G wpq1; p s 1, . . . m y 1. A scoring vector w defines a scoring rule f w as follows: each individual gives wp points for the alternative she ranks in p-th position in her individual preference, and the social ordering is determined by the total score each alternative receives over the whole population, where more is better. For three-candidate elections and the family of scoring vectors wl s Ž1, l,0., l g w0,1x, Saari demonstrates that a single profile could support up to seven different rankings 3 as the choice of l varies! As an extreme example, for 10-candidate elections, a single profile may generate over 84 millions different rankings. Although the individual preferences are unchanged, changing the scoring vector may lead to contradictory outcomes, where a candidate wins with some procedures, and is ranked in last position with others. Similar negative results are obtained when others procedures are used. Let us now consider a voting method which does not belong to the class of scoring rules, namely the method of Copeland Ž1951.. The Copeland rule is a simple and elegant extension of the majority principle, also called the Condorcet criterion. Instead completely ranking the candidates, suppose each voter compares the alternatives by pairs. For each pair of alternatives a voter will choose one, and the candidate a p will beat her opponent a q in the pairwise contest if the number of voters who prefer a p to a q is greater than the number of individuals who have the opposite preference. According to Condorcet Ž1785., the winner should be the candidate who defeats all her opponents in pairwise comparisons. Unfortunately, the following three-voter example shows a Condorcet winner may not exist for some preference profiles. Voter 1 2 3 Preference 4 a1 % a2 % a3 a 2 % a3 % a1 a 3 % a1 % a 2 In such situations, Copeland proposed to rank the candidates according to the number of victories they obtain in pairwise comparisons. In this three-candidate example, each candidate looses and wins one of her confrontations, and the Copeland outcome is a complete tie among the candidates. Saari and Merlin Ž1996. considered the question of the existence of any relationship between the Copeland outcome and the positional rankings. First, they studied its relationships with a specific scoring rule, the Borda count. In the original method described by Borda Ž1781., each voter gives m y 1 points for the 3 Notice that the result of the aggregation process may be a weak ordering, with ties between different candidates. In this case, four of them are linear orderings and three of them present tied outcomes. 4 a1 % a 2 % a 3 means that a1 is strictly preferred to a2 , and a 2 is strictly preferred to a3 . 186 V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 alternative she prefers, m y 2 for her second best choice, etc., 1 point for the next to the last and 0 point for her bottom ranked alternative. It is easy to check that the Borda count is equivalent to the positional rule described by the scoring vector w s Ž1,Ž m y 2.rŽ m y 1., . . . , Ž m y p .rŽ m y 1., . . . Ž1.rŽ m y 1.,0.. Although Saari and Merlin recover some classic relationships between the Borda and Copeland rankings when there exists a Condorcet winner, these social orderings can be as different as desired as soon as there are at least five candidates in contention and when the majority relation is cyclic. Their conclusions are even worse when they consider other scoring rules, as there might be no similarity between the f w ranking and the Copeland ordering when m G 3, whatever the state of the majority relation is. Nevertheless, even if extreme discrepancies may arise among voting rules for some preference profiles, one may wonder if these situations are just rare oddities or, on the contrary, reveals a major issue for Social Choice Theory. The difficulty of collecting complete information about the preferences of the individuals makes empirical studies, which could answer this question, quite rare. A way to circumvent this difficulty and to provide information about the occurrence of voting outcomes is to make assumptions about the distribution of the different preference types in the population. This approach leads to the definition of two major hypotheses, the Impartial Culture condition ŽIC. and the Impartial Anonymous Culture condition ŽIAC.. Both models will be described later on. Using these models and following the old Borda–Condorcet debate, there exists an important literature about the probability that a given scoring rule selects the Condorcet winner Žfor a recent survey, see Gehrlein, 1997.. These studies give the probabilities that a certain positional method agrees with any voting procedure satisfying the majority criterion Žas, for example, the Copeland method. when the Condorcet winner exists. It is proved that the Condorcet efficiency of scoring rules decreases as the number of candidates increases, and that the Borda count is generally the positional rule which has the highest probability of picking the Condorcet winner among all the positional rules Žsee van Newenhizen, 1992.. 5 Nevertheless, the existence of Condorcet winner becomes less frequent as the number of candidates increases, and the Condorcet efficiency measures become a poor indicator of the agreement between two voting rules. On the other hand, the discrepancies among positional rules have received far less attention. Two articles by Gehrlein and Fishburn Ž1983. and Saari and Tataru Ž1999. look at this issue. Both give exact computations in the case of three-candidate elections under the IC assumption when n `. The former gives the probability W Ž l, lX . that two different scoring rules, respectively, characterized ™ 5 Lepelley Ž1995; 1996. showed that these conclusions are no longer true if we assume single-peaked preferences. Tanguian Ž1996. also obtains different results assuming cardinal preferences for the voters. V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 187 by the vectors wl s Ž1, l, 0. and wlX s Ž1, lX , 0. pick the same winner as a function of l and lX : W Ž l , lX . s 1 3 q q 3 4p 3 4p 2 arcsin Ž 2 g Ž l , lX . . q arcsin Ž g Ž l , lX . . 2 arcsin Ž 2 g Ž l , lX . . y arcsin Ž g Ž l , lX . 2 . Ž 1.1 . where X g Ž l, l . s 2 y l y lX q llX 4 Ž 1 y l q l2 . 1r2 Ž 1 y lX q lX 2 . 1r2 . Ž 1.2 . The latter estimates the likelihood of obtaining one, three, five or seven different outcomes when the number of points a voter gives to her second ranked candidate varies. We also mention the work of Nurmi Ž1987.. He examines the probability that two rules select the same winner for several pairs of procedures, for finite numbers of voters Žfrom five to 301. and finite numbers of candidates Žfrom three to seven.. However, his results do not give exact figures. Except for some rare cases, exact formulas can only be obtained for three-candidate elections. The results for higher number of candidates are random simulations ŽNurmi’s case. or exhaustive countings done by computers. But, whatever the technique they use, these works conclude that discrepancies among the positional rules are not negligible. This article goes further in evaluating the discrepancies among the different voting rules. Instead of comparing only two rules, we shall evaluate the probability that most of the Õoting rules presented in the Social Choice literature select simultaneously the same winner for three-candidate elections and a large electorate under the IC assumption. More precisely, we shall consider all the rules which fulfill Condorcet criterion, all the positional methods f w and all the scoring runoff methods which use the scoring rules in a sequential elimination process. From a technical point of view, the results contained in this article are based upon recent developments in Social Choice theory. The first one concerns the analysis of scoring rules. We use properties discovered by Saari Ž1992; 1994. about the scoring rules in order to characterize the conditions under which all these methods agree. The second point concerns the techniques used for the computation of the likelihood of voting paradoxes. Until recently, the only available method under the IC hypothesis was the one developed by Gehrlein and Fishburn Ž1978a; b.. The techniques used here are developments of the ones proposed by Saari and Tataru Ž1999., and also used by Tataru and Merlin Ž1997.. They are based upon the computation of the volume of a spherical simplex and the formula of Schlafli ¨ Ž1950. Žsee also Coxeter, 1935; Kellerhals, 1989.. Nevertheless, the probability that all weighted scoring rules elect the Condorcet winner has been recovered with classical techniques by Gehrlein Ž1999.. 188 V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 The remaining of this article is organized as follows. In Section 2, we introduce our notation and describe the voting rules considered in this article. In Section 3, we characterize the conditions under which all these voting rules select the same winner for three-candidate elections. We distinguish two cases according to the existence Žor not. of a Condorcet winner. Section 4 presents the IC and the exact formulas which enable us to compute the P U value and the probabilities of several other interesting events. The description of the mathematical techniques and the detailed computations are grouped together in Section 5. 2. Notation and definitions Although the list of procedures we examine in this article is obviously incomplete, it basically includes the voting rules that are the most popular both in Social Choice literature and in real life decision processes. 6 Even if all these rules can be defined for any number of alternatives, we shall focus on the three-candidate case. Thus, there are six types of voters according to the possible preference rankings and we shall label them as follows. 1: a1 % a2 % a 3 4: a3 % a 2 % a1 2: a1 % a3 % a 2 5: a2 % a 3 % a1 3: a3 % a1 % a2 6: a2 % a1 % a3 Let n be the number of individuals, n i the number of voters of type i and call a vector n˜ s Ž n1 ,n 2 ,n 3 ,n 4 ,n 5 ,n 6 ., representing the distribution of the voters on the different preferences, a Õoting situation. By definition, Ý6is1 n i s n. Before presenting the different voting rules, we mention that we shall restrict our attention to neutral and anonymous decision processes. Neutrality means that no candidate has a particular role Žlike a status quo alternative. in the decision process, i.e., no candidate is favored. Anonymity implies the same principle for the individuals: no voter has more right than one other, and the aggregation process should only depend upon the voting situation n. ˜ 6 According to these criteria, the only important rule we do not take into account is the ApproÕal Voting, where each individual gives one point for the alternatives she approves, and zero point for the others. The advocates of approval voting generally consider dichotomic preferences: each individual is able to separate the alternatives he likes from the one he does not approve of. On the contrary, throughout this article, we shall assume that the individual preferences are linear orderings. For details about approval voting and the controversy about its qualities, see the works of Brams and Fishburn Ž1983. and Saari Ž1994., Chap. 3.4. V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 189 2.1. The Condorcet Õoting methods Let us call Condorcet voting rules the family of decision procedures which fulfill the Condorcet criterion. Denote by n p q the number of voters who prefers a p to a q minus the number of voters who have the opposite preferences. With three candidates a1 , a2 and a3 , there are three possible pairwise comparisons and we obtain for a definite voting situation n: ˜ n12 s n1 q n 2 q n 3 y n 4 y n 5 y n 6 s yn 21 n 23 s n1 y n 2 y n 3 y n 4 q n 5 q n 6 s yn 32 n 31 s yn1 y n 2 q n 3 q n 4 q n 5 y n 6 s yn13 . There are mainly two kinds of Condorcet voting rules: those which only depend upon the majority relation Žthat is the signs of the n p q ’s., and those where the size of the majorities is also a relevant data. Following Fishburn Ž1977., we shall call the former the C1 Condorcet rules, and the latter the C2 Condorcet rules. The Condorcet rules differ in the solution concept they propose to resolve a cyclic situation. A positiÕe cycle Žrespectively, a negative cycle. occurs when we observe n12 ) 0, n 23 ) 0 and n 31 ) 0 Žrespectively, n12 - 0, n 23 - 0 and n 31 - 0.. In our case, a cycle means that each candidate wins and loses one time. As we only study neutral and anonymous voting rules, all the C1 methods will select the whole set A. Hence, the Copeland method will represent all the C1 methods in this article. On the other hand, the C2 rules will use the information given by the size of the majorities in order to break the tie between the alternatives. For example, Simpson Ž1969. and Kramer Ž1977. proposed to apply the Minimax concept: the winner should be the candidate whose maximal opposition is the weakest. 7 Thus, a1 is the Minimax winner if and only if Max p Ž n p1 . is minimal among the Max p Ž n p q ., q s 1, 2, 3. We shall also consider two other famous C2 rules, the Nanson rule and the Black Method. As they are closely related to positional rules, we postpone their presentation for later. 2.2. The positional rules The scoring or positional rules have already been presented in the introduction. For the problem we consider, we shall use the scoring vector ws s Ž2, s q 1, 0., with s g wy1,1x. 8 If s s y1, we obtain one of the simplest and widely used 7 Notice that in the case of three-candidate elections, this concept of solution is equivalent to the method proposed by Kemeny Ž1959.. For more details about this C2 Condorcet method, see Fishburn Ž1977. and Young and Levenglick Ž1978.. 8 This vector is twice the vector wl s Ž1, l,0. presented in the introduction. Thus, for any preference profile, all the positional scores double, but the relative ranking of the candidates is unchanged. 190 V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 voting method, the plurality rule: each individual only votes for the alternative she prefers, and the candidate with the greatest support wins. On the other hand, when s s 1, individuals give the same number of points Žhere two. to all the candidates except the one they rank last. This defines the antiplurality or negatiÕe plurality rule. Finally, the value s s 0 gives the Borda scoring vector w 0 s Ž2,1,0.. The Borda count has a special status among the positional rules, as the Borda scores also depend upon the pairwise majority comparisons. In fact, if it would satisfy the Condorcet criterion, which is not the case, the Borda count would belong to the C2 class. More precisely, it is equivalent to use the scoring vector w s Ž1,Ž m y 2.rŽ m y 1., . . . Ž1.Ž m y 1., 0. than to compute the scores: B Ž a p ,n˜ . s Ý npq Ž 2.1 . a qgA , a q/a p and to rank the alternatives according to these scores. 9 This property lead Black Ž1958. to propose the use of Borda count in order to decide among the alternatives whenever the majority relation is cyclic. This defines a new C2 rule, namely, the Black method. 2.3. The scoring runoffs It is also possible to use the scoring rules in an elimination process. After a first stage where the alternatives are ranked with a m-dimensional scoring vector w m , let us keep the alternatives with the k 1 greatest scores, and remove from consideration the other ones. Next, a k 1-dimensional scoring vector, w k 1 , is used to rank the remaining alternatives. Then, we can stop the process at this stage or continue to remove progressively the alternatives, by using a sequence of k t-dimensional scoring vectors, 2 F k t F m, k tq1 - k t . The winner is the alternative which is ranked first with the last scoring vector. Thus, it is possible to create an infinity of scoring runoff methods, by modifying the number of steps, the number of alternatives discarded at each step, and the vectors we use Žnotice that there is no need to have any relationship among the scoring vectors used for different stages.. However, for three-candidate elections, the choice is rather limited as we can only have two stages. The first ranking is obtained with a scoring vector ws , and a pairwise comparison decides between the two remaining competitors. A variant is to keep for the second stage the alternativeŽs. whose score is superior to the average number of points a candidate receives at the first stage Žsee Kim and Roush, 1996; Lepelley and Valognes, 1999.. Hence, there is no runoff if the score of the second ranked candidates is lower that this threshold. 9 This result is common knowledge in Social Choice Theory, and it seems that it has been discovered several times. For a recent version, see the work of Saari Ž1994.. V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 191 The scoring runoffs using the Borda count have a special property, which was discovered long ago by Nanson Ž1882.. From Eq. Ž2.1., one may check that the Condorcet winner will never have a negative Borda score Žso she always obtains more than the average number of points.. Hence, she is never ranked last, and if the alternatives are removed one by one, she will be the unique winner when the Borda count is used at each step. This defines a new C2 rule, namely the Nanson rule. No other scoring runoff shares this property. 10 3. When do they agree? The references discussed in the introduction showed that the different rules we presented may end with radically different results for some definite voting situations. In this section, we focus on the conditions under which all these voting methods agree for three-candidate elections. We distinguish two cases, according to the existence or non-existence of a Condorcet winner. 3.1. The Condorcet case First, a necessary condition for all the voting rules to agree is that all the positional methods select the same winner. We shall call such a candidate an absolute positional winner. As with the Condorcet winner concept, the absolute positional winner does not exist for many voting situations. A first task is then to characterize the situation where it exists. Let qw s s Ž qw1 s,qw2 s,qw3 s . be the vector of the scores a1 , a 2 and a3 attain with the scoring vector ws for a given voting situation. Proposition 3.1. Saari Ž1992; 1994. For three-candidate elections and a giÕen Õoting situation, qw s is a conÕex combination of qw 1 and qw y 1 : qw s s 1ys ž / 2 qw y 1 q 1qs ž / 2 qw 1 . Ž 3.1 . An immediate corollary is that all the scoring rules will select the same winner if and only if the plurality and the antiplurality do. 11 Moreover, from Eq. Ž1.1., the probability that all the positional methods agree is W Ž1,0. s 0.53464, which 10 Again, this is a classical result in Social Choice Theory. We refer to Saari Ž1994. among other references. 11 This result extends to mG 3 in the following way: all scoring rules agree if the my1 positional methods defined by the vectors wk s Ž1, . . . 1,0, . . . 0., where the k first coordinates are 1 and the remaining ones 0, k g 1, . . . my14, give the same social outcome. For more details, see the work of Saari Ž1992.. 192 V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 gives an upper bound for the probability that all the voting rules agree. Also, notice that the existence of an absolute positional winner is far less frequent than the existence of a Condorcet winner Žrecall from Guibauld Ž1952. that this probability is Pc s 0.91226.. One may think that a candidate selected simultaneously by all positional rules would be an uncontested winner. Unfortunately, we know since Condorcet Ž1785. that even when they both exist, the absolute positional winner and the Condorcet winner may not coincide. Hence, knowing the likelihood that these two important concepts agree is, by itself, an interesting issue for Social Choice Theory. Proposition 3.2. The probability that the Condorcet winner and all positional rules agree for three-candidate elections is equal to the probability that both the plurality and the antiplurality rules select the Condorcet winner. MoreoÕer, all the scoring runoffs will also pick the Condorcet winner in these situations. The first part of the assertion is a direct consequence of Proposition 3.1. It is also easy to check that in these situations the scoring runoffs will choose the Condorcet winner, since an absolute winner is never removed from consideration at the first stage. U be the conditional probability that all the rules select the Condorcet Let PCon winner when it exists. For computation purposes, we shall evaluate PConŽ s .: the probability that a candidate is simultaneously the Condorcet winner, the plurality winner, and is also selected when we use the scoring vector ws , s g wy1,1x. If s s y1, this value reduces to the Condorcet efficiency of the plurality rule and from Gehrlein and Fishburn Ž1978a; b. we know it is 0.69076. If s s 0, we obtain the joint Condorcet efficiency of plurality rule and Borda count, and we should recover the value 0.65681 computed by Gehrlein Ž1998.. The value of PConŽ s . is unknown for any other s, and, by computing it for s s 1, we shall obtain a lower U bound for the probability that all the rules agree. As a corollary, PConŽ1.rPc s PCon . The fact that a candidate, a1 for example, beats, respectively, a2 and a3 when we use the scoring rule f w s is described by Eqs. Ž3.2. and Ž3.3.. Ž 1 y s . n1 q 2 n 2 q Ž s q 1 . n 3 y Ž s q 1 . n 4 y 2 n 5 q Ž s y 1 . n 6 ) 0 Ž 3.2 . 2 n1 q Ž 1 y s . n 2 q Ž s y 1 . n 3 y 2 n 4 y Ž s q 1 . n 5 q Ž s q 1 . n 6 ) 0 Ž 3.3 . Candidate a1 is also a Condorcet winner if and only if Eqs. Ž3.4. and Ž3.5. are satisfied. n1 q n 2 q n 3 y n 4 y n 5 y n 6 ) 0 n1 q n 2 y n 3 y n 4 y n 5 q n 6 ) 0. Finally, when s s y1, Eqs. Ž3.2. and Ž3.3. reduce to Eqs. Ž3.6. and Ž3.7., adds two other constraints to the set of profiles we would like to describe. n1 q n 2 y n 5 y n 6 ) 0 n1 q n 2 y n 3 y n 4 ) 0 Ž 3.4 . Ž 3.5 . which Ž 3.6 . Ž 3.7 . V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 193 Thus, the probability PConŽ s . is three times the probability that a1 is selected by all these rules, that is, three times the probability that a voting situation n˜ satisfies Eqs. Ž3.2., Ž3.3., Ž3.4., Ž3.5., Ž3.6. and Ž3.7.. 3.2. The cyclic case In some sense, the existence of a cycle shows that the preferences become quite heterogeneous. In that case, it seems unlikely to find profiles of preferences where all the rules will select the same winner. More precisely, we have to face two problems. The first one is that the different solution concepts based on the majority criterion are now free to diverge because a Condorcet winner no longer exists. The second point concerns the relationship between scoring rules and scoring runoff methods: even if a1 is top ranked for any ws , she might lose the runoff. Despite these difficulties, there still exists a small set of profiles where almost all the rules agree. Proposition 3.3. Consider a three-candidate election, where the preferences present a cycle and n12 ) 0, n 23 ) 0, n 31 ) 0. Then all the scoring runoffs and all the Condorcet rules considered in this article will select a1 if and only if all the scoring rules giÕe the same social ordering a1 % a2 % a3 . Proof. The case of the scoring runoff is simple. If a3 is second ranked for some ws , she will win the runoff while a1 is selected by all the f w s rules. For three-candidate elections, as each candidate obtains one victory and one defeat, the C1 rules select the whole set  a1 , a 2 , a 34 as a solution. Hence, they agree with any other decision process. Among the C2 voting rules, we already know that the Nanson rule will select a1 since it is a scoring runoff method, and that the Black method will also pick a1 as it uses the Borda count in case of cycles. Now, let us consider the Minimax case. If the Borda ordering is a1 % a2 % a3 , we obtain from Eq. Ž2.1. n12 q n13 ) 0 Ž 3.8 . n 31 q n 32 - 0 Ž 3.9 . because the score of the Borda winner Žrespectively, Borda loser. is greater Žrespectively, smaller. than the average number of points a candidate receives. Hence, n12 ) n 31 and n 23 ) n 31. The victory of a 3 is obtained with the smallest margin, and a1 is the Maximin winner. I In order to compute the probability that all the rules pick the same winner in the cyclic case, we shall evaluate Pcy Ž s ., the probability of the following event: 194 V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 a i % a j % a k is selected by the two scoring rules using the vectors Ž2,1 q s,0. and Ž2,1 y s,0. while n k i ) 0. For s s 0, the fact that an ordering, a1 % a2 % a 3 for example, is the Borda ranking is described by the following two equations. n1 q 2 n 2 q n 3 y n 4 y 2 n 5 y n 6 ) 0 Ž 3.10 . n1 y n 2 y 2 n 3 y n 4 q n 5 q 2 n 6 ) 0 Ž 3.11 . In the proof of Proposition 3.3, we showed that these two inequations imply that both n12 and n 23 are greater than n 31. This enables us to remove from consideration two equations, n12 ) 0 and n 23 ) 0. Thus, we just have to consider one extra constraint, n 31 ) 0, to be sure that we have a positive cycle: yn1 y n 2 q n 3 q n 4 q n 5 y n 6 ) 0. Ž 3.12 . When s ) 0, we obtain the probability that all the scoring rules f w t , t g wys, s x select the same ordering a i % a j % a k while n k i ) 0 Žsee Proposition 3.1.. The fact that the Borda rule still belongs to this interval ensures that n i j and n jk are still positive. For the ordering a1 % a2 % a 3 , this event is described by the following four equations together with Eq. Ž3.12.. Ž 1 y s . n1 q 2 n 2 q Ž 1 q s . n 3 y Ž 1 q s . n 4 y 2 n 5 y Ž 1 y s . n 6 ) 0 Ž 3.13 . 2 n1 q Ž 1 y s . n 2 y Ž 1 y s . n 3 y 2 n 4 y Ž 1 q s . n 5 q Ž 1 q s . n 6 ) 0 Ž 3.14 . Ž 1 q s . n1 q 2 n 2 q Ž 1 y s . n 3 y Ž 1 y s . n 4 y 2 n 5 y Ž 1 q s . n 6 ) 0 Ž 3.15 . 2 n1 q Ž 1 q s . n 2 y Ž 1 q s . n 3 y 2 n 4 y Ž 1 y s . n 5 q Ž 1 y s . n 6 ) 0 Ž 3.16 . Thus, the probability that all the aggregation procedures agree when the majority relation is cyclic is six times the probability that a situation n˜ satisfies Eqs. Ž3.12., Ž3.13., Ž3.14., Ž3.15. and Ž3.16.. PcyU , the conditional probability that all the rules agree when there is a positive cycle is equal to Pcy Ž1.rŽ1 y Pc .. Hence, the probability that all the rules considered in this article agree, P U , is equal to PConŽ1. q Pcy Ž1.. 4. Probability measures 4.1. The impartial culture assumption The probabilities PConŽ s . and Pcy Ž s . obviously depend on the assumptions one makes about the occurrence of the different voting situations. There are two main simplifying assumptions in the Social Choice literature about such occurrences. The first one, the IC, assumes that each voter independently selects his preference type according to a uniform probability distribution. Hence, the probability that a voter will have type i, denoted by pi , is 1r6 for i s 1, . . . , 6 in a three-candidate election. In turn, this means that the probability of having a specified voting V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 195 situation is pŽ n˜ s Ž n1 ,n 2 ,n 3 ,n 4 ,n 5 ,n 6 .. s ŽŽ n!.rŽ n1!n 2 !n 3!n 4 !n 5!n 6 !.. = 6yn . The second model, based upon the IAC assumption, regards all voting situations as equally likely, and therefore pŽ n˜ s Ž n1 ,n 2 ,n 3 ,n 4 ,n 5 ,n 6 .. s Ž n!5!.rŽŽ n q 5.!.. Throughout this study, we shall retain the IC assumption, and let n, the size of the population, tend to infinity. For more details about these assumptions and the use of probability models in voting theory, see the works of Berg Ž1985., Berg and Lepelley Ž1994. and Gehrlein Ž1997.. One of the major advantages of the IC assumption is that it is possible to use the Central Limit Theorem in order to compute the probability of a given event for a large population. Let x k be the random variable that associates to each voter k a vector of the form Ž0,0,0,1,0,0 . with probability 1r6 of having 1 in each position. Then, the expectation of x k is EŽ xk . s ž 1 1 1 1 1 1 , , , , , 6 6 6 6 6 6 / and the covariance matrix is a diagonal 6 = 6 matrix with the common entry s given by 2 s 2 s E Ž x k2 . y E Ž x k . . Let 0 n 1 T T m s Ž m1 ,m 2 , . . . m 6 . s s'n 0 n1 6 .. .. . y . n n6 6 . The Central Limit Theorem in R 5 implies the following convergence in measure: T mw m x ™ ™ ¨ Ž'2 p . e 1 5 y< t < 2 2 l as n ` where t s Ž t 1 , t 2 , . . . , t6 . g R 6 , < t < 2 s t 12 q . . . qt62 and l is Lebesgue measure on the five-dimensional hyperplane t 1 q . . . qt6 s 0. Note that since mT has the measure supported on the hyperplane m1 q . . . qm 6 s 0, the limit of mT as n ` is also a measure supported on t 1 q . . . qt6 s 0. In order to compute PConŽ s . Žrespectively, Pcy Ž s .., we shall evaluate the probability that a voting situation fulfills the set of Eqs. Ž3.2., Ž3.3., Ž3.4., Ž3.5., Ž3.6. and Ž3.7. Žrespectively, Eqs. Ž3.12., Ž3.13., Ž3.14., Ž3.15. and Ž3.16... By subtracting or dividing the number of voters of each type by the same constant, the quantities change but the comparison between them is unchanged. Therefore, one V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 196 can easily claim that n˜T satisfies conditions Ž3.2. to Ž3.7. or Ž3.12. to Ž3.16. if and only if mT satisfies these equations. The Central Limit Theorem yields ¨ Ž'2 p . H e 1 P Ž m satisfies Ž 3.2 . to Ž 3.7 . . T 5 y< t < 2 2 dl C1 where C1 s  t g R 6 ,t satisfies Eqs. Ž3.2., Ž3.3., Ž3.4., Ž3.5., Ž3.6. and Ž3.7. and Ý6is1Ž t i . s 04 . The same reasoning holds for C2 s  t g R 6 ,t satisfies Eqs. Ž3.12., Ž3.13., Ž3.14., Ž3.15. and Ž3.16. and Ý6is1Ž t i . s 04 . Because conditions Ž3.2. to Ž3.7. are homogeneous, the domain C1 of the integration is a cone. Also the measure m' y< t < 2 1 Ž '2 p . 2 e 5 l is absolutely continuous and radially symmetric. Hence, computing y1 < t < 2 1 Ž '2 p . 5 HC 2 e dl 1 reduces to finding the measure m of the cone C1 , when the measure is invariant to rotations. The measure m of such a cone is proportional to the Euclidean measure of the cone, that is, the measure on the sphere. Thus, evaluating PConŽ s . and Pcy Ž s . reduces to computing the measures of cones C1 and C2 . This goal will be achieved by using Schlafli’s formula Žsee ¨ . Coxeter, 1935; Schlafli, ¨ 1950; Kellerhals, 1989 . As the computations are rather tedious, we postpone the presentation of the Schlafli’s technique to Section 5, and ¨ we now give the results. 4.2. The results First, let us define the following quantities, I1Ž s ., I2 Ž s . and I3 Ž s .. I1 Ž s . ž 4 s qarccos sy q arccos ž' / y1 y s 13 q 2 s q 37s 2 ' Ž3 q s . 2 q 6 s ž y ž' ' y5s 7 1q 4 s2 / ž' y arccos ž qp 2 ž' / 1 q arccos 7 1 q 4 s2 3q s '3 Ž1 q s . ž yarccos ž' // 3 y arccos ' 4 2s 13 q 2 s q 37s 2 / 2 ž' '37 q 2 s q 13 s . '14 q 6 s ' '13 q 2 s q 37s 2 // 2 / q arccos ž' ' 7 y7 y s 37 q 2 s q 13 s 2 // 2 '3 Ž1 q s .arccos ž' ' 2 y q arccos Ž 3 q s q 10 s 2 . 1 q 4 s2 2 14 2 ž' y arccos Ž3 q 5 s q 2 s 2 . Ž3 q s 2 '3 y / ys 1q 4 s2 2 2 s qarccos y ž' 2 3y s 37 q 2 s q 13 s 2 ' Ž 3 q s . 13 q 2 s q 5 s ž' Ž 13 q 14 s q s 2 . 37 q 2 s q 13 s 2 Ž 3 q s . 13 q 2 s q 5 s 2 / 2 '13 q 2 s q 37s 2 // Ž 4.1 . V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 197 I2Ž s . y4 s s ž ž' p y 2 q arccos / y1 y s 13 q 2 s 2 q 37s 2 q arccos ž' ' '13 q 2 s q 37s Ž s y 3. 3 q 9 s 2 3 q s2 2 // ' Ž s2 q 3. 6 s 2 q 2 2s y ' ž ž' qarccos Ž3 s y 1 . 9 q 3 s 2 Ž 1 q 3 s 2 . Ž 37 q 2 s q 13 s 2 . / ž' y arccos Ž '3 y ž qarccos ž' / y3 14 q arccos ž' y6 42 q 14 s 2 s3 q 3 sq 7 259 q 14 s q 91 s 2 / q arccos ž' 6s 63 q 210 s 2 q 63 s 4 // . '6 s q 14 2 // 3q s2 '3 Ž 1 q s . y ž ž' qarccos Ž '3 Ž1 q s . y 3q s2 I3 Ž s . s / 37 q 2 s q 13 s 2 ž y2 3 yarccos ž 3q s2 ž q 3y s2 / 3q s2 . '2 s Ž3 y s . ž' 3 q s2 '13 q 2 s q 37s ž q 2 arccos ž' // Ž 4.2 . 6 42 q 14 s 2 // 2 Ž 3 y s 2 . Ž 3 q 7s 2 . 2 2 Ž 3 q s . Ž 6 q 14 s . ž' 2 ' / q 2 arccos ' 2 Ž 3 q s . 27 q 30 s q 3 s 20 s Ž 9 q s 2 . Ž 7 q 63 s 2 . / q arccos ž' ' s 18 q 2 s 2 3 q 28 s 2 q 9 s 4 // 4 ž' 6s Ž 3 q s 2 .Ž 21 q 63 s 2 . // ' Ž 3 q s 2 . 14 q 6 s 2 ž 4 s arccos y q arccos 2 4 s qarccos q ž // 13 q 2 s q 5 s 2 3q s 12 s yarccos ' 4 2s 13 q 2 s q 37s 2 ' . 2 6 q 2 s2 yarccos ž' y arccos 13 q 2 s q 5 s 2 ž ž' ' Ž ' / 3y s 74 q 4 s q 26 s 2 9y 9 s4 'Ž3 q s 2 .Ž 9 q s 2 .Ž 1 q 9 s 2 .Ž 3 q 9 s 2 . ' Ž 3 q s 2 . 14 q 6 s 2 / Ž 4.3 . Proposition 4.1. Consider a three-candidate election where the indiÕiduals choose independently their preference ordering according to a uniform distribution. Let PConŽ s . be the probability that the plurality rule and the f w s scoring rule select simultaneously the Condorcet winner for a large population. Then, for s g wy1,0x : PCon Ž s . s 0.69076 q 3 4p s 2 Hy1I Ž s . d s 1 Ž 4.4 . V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 198 and for any s g w0,1x : PCon Ž s . s PCon Ž 0 . q 3 4p s 2 H0 I Ž s . d s. Ž 4.5 . 2 Proposition 4.2. Consider a three-candidate election with a large electorate, where the behaÕior of the indiÕidual follows the IC assumption. Then, the probability Pcy Ž s . that the rules f w s and f w y s select simultaneously the ordering a i % a j % a k while n k i ) 0 is: 3 arccos Pcy Ž s . s '2 ž / p 3 y1q 3 2p s 2 H0 I Ž s . d s. Ž 4.6 . 3 Some values for PConŽ s . and Pcy Ž s . are displayed in Table 1. Hence, P U , the probability that all the rules select the same winner is equal to PConŽ1. q Pcy Ž1. s 0.50116. When we restrict our attention to the set of profiles where a Condorcet winner exists, this probability turns out to be 0.54751. On the other hand, the problem of the discrepancies among the different voting rules seems to be rather serious in the cyclic case, as the rules we examine agree in only 1.926% of the situations. In any case, how shall we interpret this 50% agreement level? It shows that there exists, at least in three-candidate elections, an important set of profiles where all the commonly used decision procedures agree. It is also interesting to notice that this set is different from classical preference restrictions, such as the singlepeakedness assumption Žsee Black, 1958.. Whereas single-peakedness implies the existence of a Condorcet winner and ensures a certain degree of homogeneity for the preferences Žby removing some preference types from consideration., Lepelley Table 1 s PConŽ s . PConŽ s .r Pc Pcy Ž s . Pcy Ž s .rŽ1y Pc . 1 0.8 0.6 0.4 0.2 0 y0.2 y0.4 y0.6 y0.8 y1 0.49947 0.53179 0.56606 0.60053 0.63213 0.65680 0.67216 0.68075 0.68569 0.68874 0.69076 0.54751 0.58293 0.62050 0.65829 0.69293 0.71997 0.73681 0.74622 0.75164 0.75498 0.75720 0.00169 0.00248 0.00392 0.00686 0.01367 0.03124 0.01926 0.02826 0.04470 0.07819 0.15578 0.35612 V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 199 Ž1995; 1996. showed that the scoring rules still fail to satisfy the Condorcet criterion under this condition. Moreover, Saari Ž1997. proved that they may also disagree. Thus, the next issue is to understand what is the restriction on the profiles where the voting procedures agree. On the other hand, despite the existence of this ‘island of stability’, there are always two common voting rules giving different results in at least 49.884% of the situations. If an individual Žthe planner. is free to choose the collective decision procedure, and has enough information about the individual preferences, she may affect the outcome in almost 50% of the voting situations. As we know from Social Choice literature, probabilities like P U tend to rapidly decrease as the number of candidates increases. Thus, the strategic impact of the choice of a voting procedure might become more important than the vote itself when we consider more than three alternatives. Therefore, the comparison of the voting rules and the search of the best oneŽs. remains an important task for Social Choice Theory, and a way to limit the manipulation of any individual Žor group of individuals. who has some influence on the choice of a decision procedure. 5. The Schlafli’s formula and the computation of probability under the IC ¨ assumption In order to prove Propositions 4.1 and 4.2, we have to compute volumes between hyperplanes in R 6 , i.e., the volume of cones C1 and C2 . However, this reduces to a five-dimensional problem because the measures of these cones are supported by the hyperplane t 1 q . . . t6 s 0. We refer to this equation as S1 later on. Because of the properties of the measure m , we can intersect our cones with the hypersphere in R 5 and obtain a spherical simplex, whose volume, divided by the volume of the hypersphere gives the desired probability. However, computing the volume of a four dimensional spherical simplex is not an easy task. In order to achieve this objective, we use Schlafli’s formula Žsee ¨ . Coxeter, 1935; Schlafli, 1950; Kellerhals, 1989 , which gives the differential ¨ volume of a p-dimensional spherical simplex as a function of the volume of the intersections between any two faces S j , Sk and the dihedral angle a jk between these faces: 1 dvol p Ž C . s Ž 5.1 . Ý vol Ž S l Sk . d a jk ; vol 0 s 1. Ž p y 1 . 0Fj-kFn py2 j The computation of the dihedral angles presents no difficulty. Hence, the problem basically reduces to the computation of the dihedral volumes. 5.1. Computation of PC o n(s) Let S j be the facet defined by Eq. Ž3. j .. Thus, according to Schlafli’s formula, ¨ there exists 15 possible dihedral volumes Ž S2 l S3 ., Ž S2 l S4 ., Ž S2 l S5 ., Ž S2 l S6 ., V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 200 Ž S2 l S7 ., Ž S3 l S4 ., Ž S3 l S5 ., Ž S3 l S6 ., Ž S3 l S7 ., Ž S4 l S5 ., Ž S4 l S6 ., Ž S4 l S7 ., Ž S5 l S6 ., Ž S5 l S7 . and Ž S6 l S7 .. 5.1.1. Computation of the dihedral angles ™ For each Eqs. Ž3.2., Ž3.3., Ž3.4., Ž3.5., Ž3.6. and Ž3.7., let Vj be a normal vector to the hyperplane S j . ™ ™ 5 V2 5 s 2's 2 q 3 ™ ™ V3 s Ž 2,1 y s, s y 1,y 2,y s y 1, s q 1 . , 5 V3 5 s 2's 2 q 3 ™ ™ 5 V4 5 s '6 V4 s Ž 1,1,1,y 1,y 1,y 1 . , ™ ™ 5 V5 5 s '6 V5 s Ž 1,1,y 1,y 1,y 1,1 . , ™ ™ 5 V6 5 s 2 V6 s Ž 1,1,0,0,y 1,y 1 . , ™ ™ 5 V7 5 s 2 V7 s Ž 1,1,y 1,y 1,0,0 . , ™ ™ As V and V are, respectively, normal to S and S : V2 s Ž 1 y s,2, s q 1,y s y 1,y 2, s y 1 . , j k ™™ ™ ™ V ,V s 5 V 5 P 5 V 5 P cos Ž p y a j k j k j jk k ™ ™ . s y5 V 5 P 5 V 5 P cos Ž a j k jk .. In deriving the dihedral angles, we are only interested in the angles which depend upon s. There are eight of them: a 24 s a 35 s arccos a 25 s a 34 s arccos a 26 s a 37 s arccos a 27 s a 36 s arccos ž( ž( / / y4 6 Ž s 2 q 3. y2 6 Ž s 2 q 3. ž' / ž' / sy3 2 3 q s2 sy3 4 3 q s2 . Hence: d a 24 s d a 35 s d a 25 s d a 34 s y4 s Ž s q 3 . (2 Ž 3s 2 q 1 . 2 y2 s Ž s 2 q 3 . (2 Ž 3s 2 q 7 . ds ds V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 d a 26 s d a 37 s d a 27 s d a 36 s y'3 Ž3 q s2 . 201 ds y'3 Ž 1 q s . Ž 3 q s 2 . '13 q 2 s q 5s 2 d s. Since the differential angles of the other intersections are equal to zero, we remove seven dihedral volumes from consideration. 5.1.2. The Õertices of the dihedral Õolumes In order to evaluate the surfaces of the remaining dihedral volumes, first find their vertices. A direction in R 5 is given by solving a system of four linear equations. As the cone lies in the hyperplane defined by S1 , we find the coordinates of the vertices by solving systems similar to the following. °S s 0 1 S2 s 0 S3 s 0 S4 s 0 S5 s 0 S6 ) 0 S7 ) 0 ~ ¢ The solutions of this system gives the direction P2345 . There are 15 such linear systems, but only eight of them will haÕe a solution. Moreover, we shall distinguish two cases: s F 0 and s G 0. 5.1.2.1. Case 1: s F 0. We obtain solutions for the following eight vertices. The faces defined by Eqs. Ž3.2. and Ž3.3. do not intersect, i.e., we never find a solution to the systems which contain S2 s 0 and S3 s 0 simultaneously. P2456 s Ž y1,2,y 1,y 1,2,y 1 . P3456 s Ž y1 q 5s,2 y 4 s,y s y 1,y s y 1,2 s q 2,y s y 1 . P2457 s Ž 2 y 4 s,5s y 1,y s y 1,2 s q 2,y s y 1,y s y 1 . P3457 s Ž 2,y 1,y 1,2,y 1,y 1 . P2467 s Ž 1,y 1,0,0,y 1,1 . P3467 s Ž y2,2,0,0,y 1,1 . P2567 s Ž 2,y 2,1,y 1,0,0 . P3567 s Ž y1,1,1,y 1,0,0 . V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 202 Table 2 Volumes Directions for sF 0 Directions for sG 0 S2 l S4 S2 l S5 S2 l S6 S2 l S 7 P2456 , P2456 , P2456 , P2457 , P2346 , P2356 , P2346 , P2347 , P2457 , P2457 , P2467 , P2467 , P2467 P2567 P2567 P2567 P2347 , P2357 , P2356 , P2357 , P2467 P2567 P2467 , P2567 P2467 , P2567 5.1.2.2. Case 2: s G 0. We obtain solutions for the following eight directions Žin this case, the faces defined by Eqs. Ž3.4. and Ž3.5. do not intersect.. P2346 s Ž s y 1,2,y s y 1,y s y 1,2, s y 1 . P2356 s Ž 3s y 1,2 y 2 s, s y 1,y 3s y 1,2 s q 2,y s y 1 . P2347 s Ž y2 s q 2,3s y 1,y s y 1,2 s q 2,y 3s y 1, s y 1 . P2357 s Ž 2, s y 1, s y 1,2,y s y 1,y s y 1 . P2467 s Ž 1,y 1,0,0,y 1,1 . P3467 s Ž y2,2,0,0,y 1,1 . P2567 s Ž 2,y 2,1,y 1,0,0 . P3567 s Ž y1,1,1,y 1,0,0 . 5.1.3. Computations of Õolumes on the sphere The vertices of Ž S j l Sk . are the directions Pa b c d where both j and k appear as indices. According to the situation, there will three or four of them Žsee Table 2.. By symmetry, the volume Ž S2 l Sk . equals the volume Ž S3 l Sk .. This leaves only four volumes to compute in each case. We shall now detail the computations for the volume of S2 l S4 for the case s F 0. This volume is the area of a triangle on the sphere in R 3 defined by the directions P2456 , P2457 and P2467 . By the Gauss–Bonnet theorem, the area of this triangle is equal to the sum of the angles on the surface of the triangle minus p . Let b56 , b57 and b67 be the angles on the surface of the triangle, respectively, defined by the vertices P2456 , P2457 and P2467 ; d 5 , d6 and d 7 are, respectively, the angles Hence: cos Ž b67 . s cos Ž b57 . s cos Ž b56 . s cos Ž d 5 . y cos Ž d6 . cos Ž d 7 . Ž 5.2 . sin Ž d6 . sin Ž d 7 . cos Ž d6 . y cos Ž d 5 . cos Ž d 7 . Ž 5.3 . sin Ž d 5 . sin Ž d 7 . cos Ž d 7 . y cos Ž d 5 . cos Ž d6 . sin Ž d 5 . sin Ž d6 . . Ž 5.4 . V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 203 In our case: cos Ž d 5 . s cos Ž d6 . s cos Ž d 7 . s y1 q 2 s Ž 5.5 . 2'1 y s q 4 s 2 y'3 Ž 5.6 . 2 '3 Ž 1 y 3s . Ž 5.7 . . 4'1 y s q 4 s 2 Hence, by Eqs. Ž5.2., Ž5.3., Ž5.4., Ž5.5., Ž5.6. and Ž5.7. vol py 2 Ž S2 l S4 . s b56 q b57 q b67 y p s yarccos ž' q arccos s 1q4s ž' 2 / q arccos ž' y1 y s 13 q 2 s q 37s 2 y Ž 3 q s q 10 s 2 . 1 q 4 s 2 '13 q 2 s q 37s 2 / / . When a volume Ž S j l Sk . is determined by four directions, the same technique applies, except that the area is the sum of four angles minus 2p . 5.1.4. The final formulas After several computations of this type, we are able to give the exact formula for: Ø I1Ž s . s Ý ks4,5,6,7 Ž S2 l Sk .d a 2 k , s g wy1,0x Žsee Eq. Ž4.1.. Ø I2 Ž s . s Ý ks4,5,6,7 Ž S2 l Sk .d a 2 k , s g w0, 1x Žsee Eq. Ž4.2... Between y1 and 0, the differential volume is Ž2 I1 .rŽ3. as p s 4 in the Schlafli’s ¨ formula, and as, by symmetry, the volumes Ž S2 l Sk . equal the volumes Ž S3 l Sk .. We have to multiply this number by three, and to divide it by the surface of the hypersphere in R 5, v 5 s Ž8p 2 .rŽ3.. Hence, the probability that the Condorcet winner, the plurality winner and the ws-winner are the same is given by the formulas of the Proposition 4.1. The value 0.69076, the probability that the plurality winner is the Condorcet winner, comes from Gehrlein and Fishburn Ž1978a; b.. Notice that we recover for PConŽ0. the value 0.65672 already computed by Gehrlein Ž1997.. The same reasoning applies for the case s g w0,1x. 5.2. The computation of Pc y (s) The cone C2 is defined by Eqs. Ž3.12., Ž3.13., Ž3.14., Ž3.15. and Ž3.16.. In order to simplify the notation, let us call Tj the hyperplane defined by Eq. Ž3.1. j . and T1 the hyperplane Ý6js1 t j s 0. The technique is similar to the one described in the Section 5.1, but before finding the differential volume of the cone, we shall compute Pcy Ž0.. V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 204 5.2.1. Computation of Pc y (0) The fact that the ordering a1 % a2 % a3 is chosen by the Borda rule and that n 31 ) 0 is described by Eqs. Ž3.10., Ž3.11. and Ž3.12.. These three equations define a cone C3 in a three dimensional subspace and its intersection with the sphere in R 3 gives a spherical triangle. By applying Gauss–Bonnet theorem, we find that its surface is: 2 arccos '2 ž / 3 q 2p 3 . Divide it by 4p Žthe surface of the sphere. and multiply it by six Žthe number of all possible linear orderings. to obtain the value for Pcy Ž0.: 3 arccos Pcy Ž 0 . s '2 ž / 3 p y 1 , 0.03124. Notice that it is possible to obtain the same result by using related works of Gehrlein and Fishburn Ž1980.. 5.2.2. Dihedral angles ™ For each hyperplane Tj , define the normal vector Wj : ™ ™ W3 s Ž 1 y s,2,1 q s,y 1 y s,y 2,y 1 q s . ™ W4 s Ž 1 q s,y 1,y s,y 2,y 1 q s,1 y s,2 . ™ W5 s Ž 1 q s,2,1 y s,y 1 q s,y 2,y 1 y s . ™ W s Ž 1 y s,y 1 q s,y 2,y 1 y s,1 q s,2 . . W2 s Ž y1,y 1,1,1,1,y 1 . 6 The angles between the hyperplanes are easily computed using the dot products; only eight of them depend upon s: arccos arccos arccos ž s2 y 3 s2 q 3 ž ž' / s a 35 s a 46 s2 y 3 2 Ž s 2 q 3. 2 2 6 s q 18 / s a 36 s a 45 / s a 23 s a 24 s a 25 s a 26 . V. Merlin et al.r Journal of Mathematical Economics 33 (2000) 183–207 205 Their derivatives are: d a 35 s d a 36 s d a 23 s 2'3 3 q s2 y12 s Ž 3 q s 2 . '27 q 30 s 2 q 3s 4 2s Ž 3 q s 2 . '14 q 6 s 2 . 5.2.3. Vertices of the dihedral Õolumes The intersection of the hyperplanes Ti , Tj , Tk and Tl gives the vertex Q i jk l . There are five of them. Q2345 s Ž y1 q 3s,2 y 2 s,y 1 q s,y 1 y 3s,2 q 2 s,y s y 1 . Q2346 s Ž 1 q 3s,1 y s,y 2 q 2 s,1 y 3s,1 q s,y 2 y 2 s . Q2356 s Ž 1 q 3s,y 2 y 2 s,1 q s,1 y 3s,y 2 q 2 s,1 y s . Q2456 s Ž y1 q 3s,y 1 y s,2 q 2 s,y 1 y 3s,y 1 q s,2 y 2 s . Q3456 s Ž 1,y 1,1,y 1,1,y 1 . 5.2.4. The Õolumes Due to symmetry, there are only three volumes on the sphere to compute. They are given in Table 3. Using again Gauss–Bonnet theorem, we are able to compute the surface of these spherical triangles, and we find the following equation Žsee Eq. Ž4.3..: I3 Ž s . s Ý vol Ž Tj l Tk . d a jk . js2,3 ks5,6 Finally, we have to multiply I3 Ž s . by six and divide it by the volume of the hypersphere in R 5 to obtain Žd Pcy Ž s ..rŽd s .. This completes the proof of Proposition 4.2. 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